Object Merge ShortCut
Idle Chit-chat

I have come up with a way of doing it that has many steps, but turns out a nice result.

How to Do It

Start by selecting a single vertex of your sphere in Edit Mode and keying Ctrl + L to select the rest of it. Then key P and click on Selection. This should seParate your sphere from the cube mesh. Tab into Object Mode.

Now select the cube and go to the Object modifiers tab in the Properties panel. Add a Boolean modifier. Set the settings to the following. Substitute the name of your sphere mesh for where it says ""



If you key Z to enter wireframe mode, your meshes should look something like this.



Apply the Boolean modifier by clicking the Apply button.

Make sure you are in Orthographic View (toggle with Numpad 5). Key Numpad 1, Numpad 3, or Numpad 7 until you are looking at the side of the two meshes like this.



Tab into Edit Mode, toggle Z into wireframe view, and Box select these verticies.



Key X and select Verticies. Tab into Object Mode. Key R, then Z, then 90 to rotate the hole in your cube away from the sphere.

Select the sphere and go to the Object modifiers tab in the Properties panel. Add a Boolean modifier. Set the settings to the following. Substitute the name of your cube mesh for where it says "".



If you key Z to toggle wireframe view, your meshes should look something like this.



Apply the Boolean modifier by clicking the Apply button. Select the cube and key Alt + R. The hole on the side should go back to facing the sphere.

Select the sphere mesh, then Shift + RightMouseButton select the cube mesh. Key Ctrl + J to join the two. Tab into Edit Mode. Key A once or twice to select all of both meshes, key W, and click on Remove Doubles from the popup menu.

Here is a finished .blend file with the two joined meshes.